In the last 10 years novel topological quantum matters have been discovered or fabricated in the labs. They show non-trivial topological properties on their Fermi surfaces and electron-electron correlations, leading to novel quantum phases with exotic transport and magnetic properties at surfaces/edges.This new research direction is of great current interest internationally and are of fundamental importance. In Taiwan, there is growing interest along these directions both theoretically and experimentally. This TG has accomplished significant research breakthrough during 2014-2018 on this topic (see our progress report). There are still many important un-solved and new topics yet to be addressed. Our TG for 2019-2020 continues along this line with special emphases on (i) the interplay between correlation and topology in unconventional superconductivity, which shows significance in realizing new class of topological and unconventional superconductors and (ii) signatures and correlation effects of Majorana fermions/Majorana zero-modes at edges of topological superconductors, which shows significance in quantum computation and quantum information.

Goals:

The purpose of forming this TG is to join our efforts and establish internationally competitive domestic

research collaborations on the fundamental issues of these topics, and to foster our young theorists. The

main topics include:

Unconventional superconductivity in topological materials

(CY Mou + CH Chung + DW Wang + SM Huang + TG collaborators)

One of most important correlation effects in topological quantum matter is the emergence of unconventional superconductivity. In this subject, we shall aim at fostering collaborations with experimentalists and explore unconventional pairing symmetries that arise from topological effects. It is proposed that the interplay between spin-orbit interaction, Coulomb interaction and phononmediated interaction, together with geometrical frustration, may lead to novel superconducting phases. Specifically, we shall explore superconductivity due to inter-surface states pairing in topological insulators, superconductivity involved in Dirac/Weyl semimetals, and possibility of high temperature superconductivity in flat bands that are generated by topological effects. The flat-band superconductivity has been recently observed in bi-layer graphene[1,2]. We will identify systems with flat-bands generated by topology and study the underlying superconductivity arising from the Coulomb interaction. We will also study the possibility of superconducting stability with pair-density-wave states in flat-bands and Weyl semimetals These novel superconducting phases are hard to be realized in conventional materials and could be topological in nature and host Majorana fermions more easily that the conventional materials do.

In condensed matter systems, a Majorana fermion (MF) can appear as a localized edge state of a topological superconductor. It is not that surprising because it is known that Bogoliuov quasi-particle is indeed a linear combination between the ordinary particle (electron) and its anti-particle (hole). When the MFs are localized at the edge of a TO phase, they are also called Majorana zero mode (MZM), which may be used for topological quantum computation. Certain experimental signature has been proposed in an ordinary s-wave superconducting wire with strong spin-orbital coupling through proximity effect or even in systems of ultracold atoms.

However, besides of the single particle properties of a MF, in this proposal, it is also a very important question regarding to its many-body effects, when the mutual interaction is considered. This is especially important when MFs are generated not only locally but also macroscopically. Therefore, it can be a completely new direction and new research area to study how these non-Abelian particles can interact with each other and what the ground state it can be. The classification of these interacting topological orders is still not very well understood yet, and will also an important direction for our future study. For example, exotic new type of spin liquid state may emerge out of interacting Majorana zero modes made of Cooper boxes [1].

Besides of some analytic and numerical study on these interesting systems, it also has been proposed that recent rapid development on Deep Learning (DL) may be applied to the investigation of quantum many-body physics. One possible direction is to apply DL to learn the physical quantities calculated from some exactly solvable model via different parameter regime. The trained DL model maybe then used for the classification of some other unresolved model, which may show their topological properties from these physical quantities, even the full theory may be still unclear. We could have a close collaboration with TG4 to develop codes on this direction.

Recently, there has been intense interest in the symmetry protected topological phase transition in Fermion parity. An elegant system that pertains to such phase transitions involves a nonsuperconducting material sandwiched between two superconductors with possibly a nonzero relative phase. One of the important underlying symmetries is the so-called particle-hole symmetry. Therefore, by tuning some external parameters, the parity of the ground state may change. Here we propose to study such a system by including spin-orbit interaction in the non-superconducting material, e.g., semiconductors. By introducing an external magnetic field as the external parameter, the Josephson junction setup can also possess 0-\pi phase transition in addition to the Fermion parity switch. Here, we wish to disentangle the connection between 0-\pi phase transition and the Fermion-parity switch. Our theoretical approach is based on solving the Bogoliubov-de Gennes equation numerically. This approach has been successfully developed and adopted by our group in the past. We expect that our results on these systems may have important applications in spintronics as well as quantum computations.

Novel theoretical approach to correlated electron systems

(SM Huang + CT Wu and TG collaborators)

To solve a strongly interacting system is quite a tough task. To make progress, we have to do some approximations. There are some methods to solve the Hubbard model; for example, by the so-called Gutzwiller approximation or by dynamical mean-field theory. However, these methods though include on-site correlation or temporal quantum fluctuation, do not reveal correct spatial correlations. So we should extend the on-site approximation to the cluster approximation. We would like to build up a solvable effective Hamiltonian in the spirit of cluster perturbation theory. The model will be constructed on the basis of exact many-body eigenstates of finite-size clusters. The effective model of plenary coupling parameters will be tested as a good approximation through the machine learning scheme. The effective Hamiltonian will provide us a more credible base which eigenstates certainly exhibit quantum entanglement. This approach is relevant for addressing topological physics in correlated electron systems.

Superconductors with broken time-reversal and chiral symmetries have received a lot of attention. With the recent interest in topological superconductors, a much discuss topic is the spontaneous (at zero magnetic field) thermal Hall conductivity of these superconductors, where edge states would contribute a universal thermal Hall conductance at low temperatures (e.g [1,2]). However, almost all candidates of superconductors with the appropriate broken symmetries exhibit behavior (such as thermal conductivity) that can be understood only if there exist nodes in their energy gaps. In this case, since impurity are pair-breaking, it is expected that impurity generated unpaired carriers would also contribute to the thermal Hall conductivity as well as the diagonal thermal conductivity. Recently, one of us [3] investigated just such an impurity mechanism of generating the thermal Hall effect for Sr2RuO4, where the order parameter is supposed to be chiral p-wave. It is demonstrated that the impurity mechanism yields in general a much larger contribution (for realistic samples) than from the topological edge state contributions examined by other authors [1].

We plan to extend our investigation along these lines. For example, another superconductor with broken time and chiral symmetry is UPt3, which has a different order parameter from Sr2RuO4. There is a proposal that the topological edge states contribute a universal thermal Hall conductivity in this system [2]. We would examine whether the impurity contribution would again potentially provide a larger contribution. It is simple to show that, for the popular chiral f-wave order parameter examined in [2], isotropic, random, uncorrelated impurities would not generate the corresponding effect of [3]. However, it is entirely conceivable a more refined model would do and be relevant experimentally for realistic systems. We plan to examine this question.

We believe that some of these results can be published in top journals, and some of our members may be able to present our results in international workshops/conferences to enhance our international visibility.

We also expect to establish long-term important international collaborations (see possible collaborators above). Meanwhile, we expect to achieve domestic collaborations between theorists and experimentalists as well as between young participants (PhD students and postdocs) and core members.